A colored operad for string link infection

Abstract

Budney constructed an operad that encodes splicing of knots and further showed that the space of (long) knots is generated over this splicing operad by the space of torus knots and hyperbolic knots. This generalized the satellite decomposition of knots from isotopy classes to the level of the space of knots. Infection by string links is a generalization of splicing from knots to links. We construct a colored operad that encodes string link infection. We prove that a certain subspace of the space of 2-component string links is generated over a suboperad of our operad by its subspace of prime links. This generalizes a result from joint work with Blair from isotopy classes of string links to the space of string links. Furthermore, all the relations in the monoid of 2-string links (as determined in our joint work with Blair) are captured by our infection operad.

This paper concerns operations on knots and links, particularly infection by string links. Classically, knots and links are considered as isotopy classes of embeddings of a 1-manifold into a 3-manifold, such as R3, D3, or S3. Instead of considering just isotopy classes, we consider the whole space of links, that is the space of embeddings of a certain 1-manifold into a certain 3-manifold.
We also consider spaces parametrizing the operations and organize all of these spaces via the concept of an operad (or colored operad). The operad framework is in turn convenient for studying spaces of links and generalizing statements about isotopy classes to the space level. Finding such statements to generalize was the motivation for recent work of the authors and R Blair on isotopy classes of string links [1].

Our work closely follows the work of Budney.
Budney first showed that the little 2-cubes operad C2 acts on the space K of (long) knots, which implies the well known commutativity of connect-sum of knots on isotopy classes. He showed that K is freely generated over C2 by the space P of prime knots, generalizing the prime decomposition of knots of Schubert from isotopy classes to the level of the space of knots [2]. Later, he constructed a splicing operad SP which encodes splicing of knots. He showed that K is freely generated over a certain suboperad of SP by the subspace of torus and hyperbolic knots, thus generalizing the satellite decomposition of knots from isotopy classes to the space level [4].

Infection by string links is a generalization of splicing from knots to links. This operation is most commonly used in studying knot concordance. One instance where string link infection arises is in the clasper surgery of Habiro [15], which is related to finite-type invariants of knots and links. In another vein, Cochran, Harvey, and Leidy observed that iterating the infection operation gives rise to a fractal-like structure [9]. This motivated our work, and we provide another perspective on the structure arising from string link infection. We do this by constructing a colored operad which encodes this infection operation. We then prove a statement that decomposes part of the space of 2-component string links via our colored operad.

Splicing and infection are both generalizations of the connect-sum operation. The latter is always a well defined operation on isotopy classes of knots, but if one considers long knots, it is even well defined on the knots themselves. This connect-sum operation (i.e., “stacking”) is also well defined for long (a.k.a. string) links with any number of components. Thus we restrict our attention to string links.

1.1 Basic definitions and remarks

Let I=[−1,1] and let D2⊂R2≅C be the unit disk with boundary.

Definition 1.1.

whose values and derivatives of all orders at the boundary points agree with those of a fixed embedding ic.
For concreteness, we take ic to be the map which on the ith copy of I is given by t↦(t,xi)
where xi=(i−1c,0). We will call ic the trivial string link.
Another example of a string link is shown in Figure 1.

D2×{0}D2×{1}Figure 1: A string link

In our work [1], our definition of string links allowed more general embeddings, and the ones defined above were called “pure string links.” We choose the definition above in this paper because infection by string links behaves more nicely with this more restrictive notion of string link. (Specifically, it preserves the number of components in the infected link.)

The condition on derivatives is not always required in the literature.1 We impose it because this allows us to identify a c-string link with an embedding ∐cR↪R×D2 which agrees with a fixed embedding outside of I×D2. Let Lc=Emb(∐cR,R×D2) denote the space of c-string links, equipped with the C∞ Whitney topology. An isotopy of string links is a path in this space, so the path components
of Lc are precisely the isotopy classes of c-string links. Often we will write K for the space L1 of long knots.

The braids which qualify as string links under Definition 1.1 are precisely the pure braids.
There is a map from Lc to the space Emb(∐cS1,R3) of closed links in R3 by taking the closure of a string link. When c=1, this map is an isomorphism on π0. In other words, isotopy classes of long knots correspond to isotopy classes of closed knots.
In general, this map is easily seen to be surjective on π0, but it is not injective on π0. For example, any string link and its conjugation by a pure braid yield isotopic closed links, and for c≥3, there are conjugations of string links by braids which are not isotopic to the original string link.
We will sometimes write just “link” rather than “string link” or “closed link” when the type of link is either clear from the context or unimportant.

1.2 Main results

Our first main result is the construction of a colored operad encoding string link infection. An operad O consists of spaces O(n) of n-ary operations for all n∈N. Roughly, an operad acts on a space X if each O(n) can parametrize ways of multiplying n elements in X. (We provide thorough definitions in Section 3.) A colored operad arises when different types of inputs must be treated differently. In our case, we have to treat string links with different numbers of components differently, so the colors in our colored operad are the natural numbers. This theorem is proven as Theorem 5.6 and Proposition 6.3.

Theorem 1.

There is a colored operad I which encodes the infection operation and acts on spaces of string links Lc for c=1,2,3,....

When restricting to the color 1, the (ordinary) operad I{1} which we recover is Budney’s splicing operad, and the action of I{1} on K is the same as Budney’s splicing operad action.

For any c, the operad I{c} obtained by restricting to c is an operad which admits a map from the little intervals operad C1. The resulting C1-action on Lc encodes the operation of stacking string links.

On the level of π0, our infection operad encodes all the relations in the whole 2-string link monoid.

We then use our colored operad to decompose part of the space of string links. We rely on an analogue of prime decomposition for 2-string links proven in our joint work with R Blair [1], so we must restrict to c=2.
We consider a “stacking operad” I#, which is a suboperad of I{2} and which is homeomorphic to the little intervals operad. This operad simply encodes the operation of stacking 2-string links in I×D2, with the little intervals acting in the I factor.
The theorem below is proven as Theorem 6.8.

Theorem 2.

Let π0S2 denote the submonoid of π0L2 generated by those prime 2-string links which are not central. (By [1], this monoid is free.) Let S2 be the subspace of L2 consisting of the path components of L2 that are in π0S2.
Then π0S2 is freely generated as a monoid over the stacking suboperad I#;
The generating space is the subspace consisting of those components in S2 which correspond to prime string links.

1.3 Organization of the paper

In Section 3, we review the definitions of an operad and the particular example of the little cubes operad. We then give the more general definition of a colored operad.

In Section 4, we review Budney’s operad actions on the space of knots. This includes his action of the little 2-cubes operad, as well as the action of his splicing operad.

In Section 5, we define our colored operad for infection and prove Theorem 1. We make some remarks about our operad related to pure braids and rational tangles, and we briefly discuss a generalization to embedding spaces of more general manifolds.

In Section 6, we focus on the space of 2-string links. We prove Theorem 2, which decomposes part of the space of 2-string links in terms of a suboperad of our infection colored operad. We conclude with several other statements about the homotopy type of certain components of the space of 2-string links.

Notation:

∐cX means X⊔...⊔Xc times

f|A means the restriction of f to A

¯¯¯¯¯X denotes the closure of X; ∘X denotes the interior of X

[a] denotes the equivalence class represented by an element a; [a1,..an] denotes the equivalence class of a tuple (a1,...,an).

1.4 Acknowledgments

The authors thank Tom Goodwillie for useful comments and conversations. They thank Ryan Budney for useful explanations and especially for his work which inspired this project. They thank Ryan Blair for useful conversations and for invaluable contributions in their joint work with him, on which Theorem 2 depends. They thank a referee for a careful reading of the paper and useful comments. They thank Connie Leidy for suggesting the rough idea of this project. They thank Slava Krushkal for suggesting terminology and for pointing out the work of Habiro. Finally, they thank David White for introducing the authors to each other. The second author was supported partly by NSF grant DMS-1004610 and partly by a PIMS Postdoctoral Fellowship.

Infection is an operation which takes a link with additional decoration together with a string link and produces a link. This operation is a generalization of splicing which in turn is a generalization of the connect-sum operation. Infection has been called multi-infection by Cochran, Friedl, and Teichner [8], infection by a string link by Cochran [7] and tangle sum by Cochran and Orr [10]. Special cases of this construction have been used extensively since the late 1970’s, for example in the work of Gilmer [14]; Livingston [22]; Cochran, Orr, and Teichner [11, 12]; Harvey [16]; and Cimasoni [6]. The operad we define in this paper will encode a slightly more general operation than the infection operation that has been defined in previous literature. This section is meant to
inform the reader of the definition in previous literature and
provide motivation for the infection operad.

2.1 Splicing

Consider a link R∈S3 and a closed curve η∈S3∖R such that η bounds an embedded disk in S3 (η is unknoted in S3) which intersects the link components transversely. Given a knot K, one can create a new link Rη(K), with the same number of components as R, called the result of splicing R by K at η. Informally, the splicing process is defined by taking the disk in S3 bounded by η; cutting R along the disk; grabbing the cut strands; tying them into the knot K (with no twisting among the strands) and regluing. The result of splicing given a particular R, η and K is show in Figure 2. Note that if η simply linked one strand of R then the result of the splicing would be isotopic to the connect-sum of R and K.

RηKRη(K)Figure 2: An example of the splicing operation.

Formally, Rη(K) is arrived at by first removing a tubular neighborhood, N(η), of η from
S3. Note S3∖N(η)⊂S3 is a solid torus with R embedded in its interior. Let CK denote the complement in S3 of a tubular neighborhood of K.
Since the boundary of CK is also a torus, one can identify these two manifolds along their boundary.
In order to specify the identification, we use the terminology of meridians and longitudes.
Recall that the meridian of a knot is the simple closed curve, up to ambient isotopy, on the boundary of the complement of the knot which bounds a disk in the closure of the tubular neighborhood of the knot and has +1 linking number with the knot. Also recall that the longitude of a knot is the simple closed curve, up to ambient isotopy, on the boundary of the complement of the knot which has +1 intersection number with the meridian of the knot and has zero linking number with the knot.

The gluing of S3∖N(η) to CK is done so that
the meridian of the boundary of S3∖N(η)
is identified with the meridian of K in the boundary of CK. Note that this process describes a Dehn surgery with surgery coefficient ∞ along K⊂S3 where the solid torus glued in is S3∖N(η). Thus, the resulting manifold will be a 3-sphere with a subset of disjoint embedded circles whose union is Rη(K) (the image of R under this identification).
Although the embedding of Rη(K) in S3 depends on the identification of the surgered 3-manifold with S3, its isotopy class is independent of this choice of identification.

2.2 String link infection

Although there is a well studied generalization of the connect-sum operation from closed knots to closed links, there is no generalization of splicing by a closed link. There is, however, a generalization of splicing called infection by a string link, which we will now define. See the work of Cochran, Friedl, and Teichner [8, Section 2.2] for a thorough reference.

By an r-multi-diskD we mean the oriented disk D2 together with r ordered embedded open disks D1,…Dr (see Figure 3). Given a link L⊂S3 we say that an embedding φ:D→S3 of an r-multi-disk into S3 is proper if the image of the multi-disk, denoted by D, intersects the link components transversely and only in the images of the disks D1,…Dr as in Figure 3. We will refer to the image of the boundary curves of φ(D1),…,φ(Dr) by η1,…,ηr.

DD1D2DrDη1ηrFigure 3: An r-multi-disk and a properly embedded multi-disk

Suppose R⊂S3 is link, D⊂S3 is the image of a properly embedded r-multi-disk, and L is an r component string link. Then informally,
the infection of R by L at D, denoted by RD(L), is the link obtained by tying the r collections of strands of R that intersect the disks φ(D1),…,φ(Dr) into the pattern of the string link L, where the strands linked by ηi are identified with the ith component of L, such that the ith collection of strands are parallel copies of the ith component of L. Figure 5 shows an example of this operation.

We now define this operation formally.
Given a string link L:∐rI↪I×D2, let CL denote the complement of a tubular neighborhood of (the image of) L in I×D2. In Figure 4 an example of a string link is shown with its complement to the right. The meridian of a component of a string link is the simple closed curve, up to ambient isotopy, on the ∂D2×I boundary of the closure of the tubular neighborhood of the component which bounds a disk and has +1 linking number with the component. We call the set of such meridians the meridians of the string link. The longitude of a component of a string link is a properly embedded line segment f:I→∂D2×I, up to ambient isotopy, on the ∂D2×I boundary of the closure of the tubular neighborhood of the component; it is required to have +1 intersection number with the meridian of that component, to have zero linking number with that component, and to satisfy f(0)=(1,0)∈∂D2×{0} and f(1)=(1,1)∈∂D2×{1}. We call the set of such longitudes the longitudes of the string link. In Figure 4 the meridians, μi, and longitudes, ℓi, are shown on the boundary of the complement. Note that the boundary of the complement of any r-component string link is homeomorphic to a genus-r orientable surface.

\footnotesizeℓ1\footnotesizeμ1\footnotesizeμ2\footnotesizeℓ2

Figure 4: A string link and its complement.

Let R⊂S3 be a link, and let L:∐rI↪I×D2 be a string link.
Fix a proper embedding of a thickened r-multidisk D×I in S3∖R.
Formally the infection of R by L at D is obtained by removing (D∖⊔iφ(D1))×I from S3 and gluing in the complement of L. Note that (D∖⊔iφ(Di))×I is the complement of a r-component trivial string link T (see Figure 5), and thus the boundary of S3∖((D∖⊔iφ(D1))×I) is a genus-r orientable surface. One identifies this boundary and the boundary of the complement of L, ∂CL, first by identifying ∂D×I with ∂D2×I subset of the boundary of CL where ∂D2×I is taken to be a subset of the boundary of D2×I where L lives, (D∖⊔iφ(Di))×{0,1} is identified with (D2∖N(L))×{0,1} and the D2×I components of the closure of N(T) and N(L) are identified so that the meridians and longitudes of L are identified with the meridians and longitudes of T.

We claim that the resulting manifold is S3 containing a link RD(L) (the image of R under this identification). The resulting manifold is homeomorphic to S3 because

S3∖Int((D∖⊔iφ(D1))×I)∪(D2×I)∖N(L)

=

(S3∖D×I)∪((⊔i(φ(Di)×I))∪(D2×I)∖N(L))

≅

S3

where the last homeomorphism follows form the observation that the previous space is the union of two 3-balls.
Again, the specific embedding of RD(L) will depend on the choice of homeomorphism, but all choices will yield isotopic embeddings.

RDFigure 5: Infection of the string link R along D by the string link L from Figure 4.

We start by reviewing the definitions of an operad O(={O(n)}n∈N), and
an action of O on X
(a.k.a. an algebra X over O). We then proceed to colored operads. Technically, the definition of a colored operad subsumes the definition of an ordinary operad, but for ease of readability, we first present ordinary operads. Readers familiar with these concepts may safely skip this Section.

3.1 Operads

Operads can be defined in any symmetric monoidal category, but we will only consider the category of topological spaces. In this case, the rough idea is as follows. An algebra X over an operad O is a space with a multiplication X×X→X, and the space O(n) parametrizes ways of multiplying n elements of X, i.e., maps Xn→X. In other words, O(n) captures homotopies between different ways of multiplying the elements, as well as homotopies between these homotopies, etc. Thus an element of O(n) is an operation with n inputs and one output. This can be visualized as a tree with n leaves and a root, and in fact, free operads are certain spaces of decorated trees.
For a more detailed introduction, the reader may wish to consult the book of Markl, Shnider, and Stasheff [23], May’s book [24], or the expository paper of McClure and Smith [25].

Definition 3.1.

An operad O (in the category of spaces) consists of

a space O(n) for each n=1,2,... with an action of the symmetric group Σn

structure maps

O(n)×O(k1)×...×O(kn)→O(k1+...+kn)

(1)

such that the following three conditions are satisfied:

Associativity: the following diagram commutes:

Symmetry:
Let σ×σ denote the diagonal action on the product O(n)×(O(k1)×...×O(kn))
coming from the actions of Σn on O(n) and on O(k1)×...×O(kn) by permuting the factors. For a partition →k=(k1,...,kn) of a natural number k1+...+kn, let σ→k∈Σk1+...+kn denote the “block permutation” induced by σ and the partition →k.

We also require that for τi∈Σki for i=1,...,n, the following diagram commutes:

Identity:
There exists an element 1∈O(1) (i.e., a map ∗→O(1)) which induces the identity on O(k) via

O(1)×O(k)

→O(k)

(1,L)

↦L

and which induces the identity on O(n) via

O(n)×O(1)×O(1)×...×O(1)

→O(n)

(L,1,1,...,1)

↦L.

∎

Some authors define the structure maps via ∘i operations, i.e., plugging in just one operation into the ith input, as opposed to n operations into all n inputs. These ∘i maps can be recovered from the above definition by setting kj=1 for all j≠i and using the identity element in O(1).

Definition 3.2.

Given an operad O, an
action of O on X
(also called an algebra X over O) is a space X together with maps

O(n)×Xn→X

such that the following conditions are satisfied:

Associativity:
The following diagram commutes:

Symmetry:
For each n, the action map is Σn-invariant, where Σn acts on O(n) by definition, on Xn by permuting the factors, and on the product diagonally. In other words, the action map descends to a map

O(n)×ΣnXn→X

Identity:
The identity element 1∈O(1) together with the map

O(1)×X→X

induce the identity map on X, i.e., the map takes (1,x)↦x.

∎

3.2 The little cubes operad

Our infection colored operad extends Budney’s splicing operad, which in turn was an extension of Budney’s action of the little 2-cubes operad on the space of long knots. Thus the little 2-cubes operad is of interest here.

Definition 3.3.

The little j-cubes operadCj is the operad with Cj(n) the space of maps

(L1,...,Ln):∐nIj↪Ij

which are embeddings when restricted to the interiors of the Ij and which are increasing affine-linear maps in each coordinate. The structure maps are given by composition:

Cj(n)×Cj(k1)×...×Cj(kn)

→Cj(k1+...+kn)

(L1,...,Ln),(L11,...,L1k1),...,(Ln1,...,Lnkn)

↦(L1∘(L11,...,L1k1),...,Ln∘(Ln1,...,Lnkn))

∎

Note that for all j≥2, the multiplication induced by choosing (any) element in Cj(2) is commutative up to homotopy, which can be seen via the same picture that shows that πjX is abelian for j≥2.

3.3 Colored Operads

Now we present the precise definitions of a colored operad and an action of a colored operad on a space. This generalization of an operad is necessary to generalize Budney’s operad from splicing of knots to infection by links.

Definition 3.4.

A colored operadO=(O,C) (in the category of spaces) consists of

a set of colors C

a space O(c1,...,cn;c) for each (n+1)-tuple (c1,...,cn,c)∈C together with compatible maps O(c1,...,cn;c)→O(cσ(1),...,cσ(n);c) for each σ∈Σn

Associativity: The map below is the same regardless of whether one first applies the structure maps to the first two factors or the last two factors on the left-hand side:

Symmetry:
The following diagram below commutes. The vertical map is induced by σ in both the first factor and the last n factors, and σ→k∈Σk1+...+kn is the block permutation induced by σ and the partition (k1,...,kn).

We also require that, for τi∈Σki, i=1,...,n, the following diagram commutes:

Identity: For every c∈C, there is an element 1c∈O(c;c) which together with

O(c;c)×O(c1,...,cn;c)→O(c1,...,cn;c)

induces the identity map on O(c1,...,cn;c). We also require that the elements 1c1,...,1cn together with

O(c1,...,cn;c)×O(c1;c1)×...×O(cn;cn)→O(c1,...,cn;c)

induce the identity map on O(c1,...,cn;c).

∎

The colors c1,..,cn can be thought of as the colors of the inputs, while c is the color of the output. A colored operad with C={c} is just an operad, where
O(c,...,cn times;c)
is O(n). Sometimes, for brevity, we write “operad” to mean “colored operad.”

Note that if we have a colored operad O with colors C and a subset C′⊂C, we can restrict to another colored operad OC′ consisting of just the spaces O(c1,...,cn;c) with ci,c∈C′ (and the same structure maps as O).

Definition 3.5.

Given a colored operad O=(O,C),
an action of O on A
(also called an O-algebraA) consists of a collection of spaces {Ac}c∈C together with maps

O(c1,...,cn;c)×Ac1×...×Acn→Ac

(2)

satisfying the following conditions:

Associativity: The following diagram commutes:

Symmetry: For each σ∈Σn, the following diagram commutes, where the vertical map is induced by the Σn-action and permuting the factors of A:

Identity:
The map induced by 1c∈O(c,c) together with O(c;c)×Ac→Ac is the identity on Ac.

∎

If we have a subset C′⊂C, the restriction colored operad OC′ acts on the collection of spaces {Ac}c∈C′.

Example 3.6.

A planar algebra as in the work of Jones [20] is an algebra over a certain colored operad. In fact, planar diagrams form a colored operad called the planar operadP. The colors C are the even natural numbers, and P(c1,...,cn;c) is the space of diagrams with n holes, ci strands incident to the i-th boundary circle, and c strands incident to the outer boundary circle. If Ac denotes the space of tangle diagrams in D2 with c endpoints on ∂D2, then the collection {Ac}c∈C is an example of an algebra over P (a.k.a. a planar algebra).

4.1 Budney’s 2-cubes action

The operation of connect-sum of knots is always well defined on isotopy classes of knots. If one considers long knots, one can further define connect-sum (or stacking) of knots themselves, rather than just the isotopy classes. That is, there is a well defined map

#:K×K→K

where K=Emb(R,R×D2) is the space of long knots. If one descends to isotopy classes, this operation is commutative, i.e., # is homotopy-commutative. See Budney’s paper [2, p. 4, Figure 2] for a beautiful picture of the homotopies involved. This picture suggests that one can parametrize the operation # by S1≃C2(2). Thus it suggests that the little 2-cubes operad C2 acts on K.

Budney succeeded in constructing such a 2-cubes action, but to do so, he had to consider a space of fat long knots

EC(1,D2):={f:R1×D2↪R1×D2|supp(f)⊂I×D2}

where supp(f) is defined as the closure of {x∈R1×D2|f(x)≠x}.
The notation EC(1,D2) stands for (self-)embeddings of R1×D2 with cubical support.
This space is equivalent to the space of framed long knots, but one can restrict to the subspace where the linking number of the curves f|R×(0,0) and f|R×(0,1) is zero; this subspace is then equivalent to the space of long knots.

The advantage of EC(1,D2) is that one can compose elements. In the 2-cubes action on this space, the first coordinate of a cube acts on the R factor in R×D2, while the second factor dictates the order of composition of embeddings. Precisely, the action is defined as follows. For one little cube L, let Ly be the embedding I↪I given by projecting to the last factor. Let Lx be the embedding I↪I given by projecting to the first factor(s). Let σ∈Σn be a permutation (thought of as an ordering of {1,...,n}) such that Lyσ(1)(0)≤...≤Lyσ(n)(0). The action

4.2 The splicing operad

In the above 2-cubes action, the second coordinate is only used to order the embeddings. Thus instead of the 2-cubes operad, one could consider an operad of “overlapping intervals” C′1. An element in C′1(n) is n intervals in the unit interval, not necessarily disjoint, but with an order dictating which interval is above the other when two intervals do overlap. Precisely, an element of C′1(n) is an equivalence class (L1,...,Ln,σ) where each Li is an embedding I↪I and where σ∈Σn. Elements (L1,...,Ln,σ) and (L′1,...,L′n,σ′) are equivalent if Li=L′i for all i and if whenever Li and Lj intersect, σ−1(i)≤σ−1(j)⇔(σ′)−1(i)≤(σ′)−1(j). It is not hard to see what the structure maps for the operad are (and they are given in Budney’s paper [4]). Budney then easily recasts his 2-cubes action as an action of the overlapping intervals operad C′1.

The splicing operad SCD21 (which we abbreviate for now as SC) is formally similar to the overlapping intervals operad, in that SC(n) consists of equivalence classes of elements (L0,L1,...,Ln,σ) with the same equivalence relation as for C′1. In the splicing operad, however, L0 is in EC(1,D2), L1,...,Ln are embeddings Li:I×D2↪I×D2, and all the Li are required to satisfy a “continuity constraint,” as follows. One considers σ∈Σn as an element of Σn+1=Aut{0,....,n} which fixes 0. If σ−1(i)<σ−1(k) one can think of Li as inner (or first in order of composition) with respect to Lk. One wants the “round boundary” of Lk not to touch Li, but for the operad to have an identity element, one needs to allow for Lk to be flush around Li. The precise requirement needed is that for 0≤σ−1(i)<σ−1(k)

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯imLi∖imLk∩Lk(∘I×∂D2)=∅.

Note that SC is a much “bigger” operad than C′1. One can think of L0 as the “starting (thickened long) knot” for the splicing operation and of the other Li as n “hockey pucks” with which one grabs L0 and ties up into n knots. This gives a map

SC(n)×Kn→K

which will define the action of the splicing operad on K. To fully construct SC as an operad, one needs the operad structure maps, which also come from the map above. Roughly speaking, the structure maps are as follows. Given one splicing diagram with n pucks and n other splicing diagrams each with ki pucks (i=1,...,n), put the ith splicing diagram into the ith puck by composing the “starting knots” L0 and “taking the pucks along for the ride.” For a precise definition and pictures, the reader may either consult [4] or read the next Section, which closely follows Budney’s construction and subsumes the splicing operad.

Definition 5.1.

We will be more concerned with this image of the fixed trivial fat string link rather than the embedding itself.

A convenient way of choosing an ic is to fix an embedding ∐cD2↪D2 and then take the product with the identity map on I. For c≥2, we choose an embedding which takes the centers of the cD2’s to the points x1,...,xc from our definition (1.1) of string links. Beyond that, we remain agnostic about this fixed embedding. For c=1, we choose i1 to be the identity map. This will recover Budney’s splicing operad from our colored operad when all the colors are 1.

Now we define the space of c-component fat string links to be

FSLc:={f:∐cI×D2↪I×D2|f agrees with ic on ∂I×D2}.

These are the spaces on which the infection colored operad will act.
An element of FSL3 is displayed in Figure 6. By our condition on the fixed trivial fat string link, we can restrict f to the cores of the solid cylinders to obtain an ordinary string link f|I×{x1,...,xc} as in Definition 1.1.

Figure 6: A fat string link, or more precisely, an element of FSL3.

5.1 The definition of the infection colored operad

We now define our colored operad I=(I,C). We put C=N+, so each color c is a positive natural number.

Definition 5.2 (The spaces in the colored operad I).

An infection diagram is a tuple (L0,L1,...,Ln,σ) with L0∈FSLc, σ∈Σn, and Li an embedding Li:I×D2↪I×D2 (for i=1,...,n) satisfying a certain continuity constraint. The constraint is that if 0≤σ−1(i)<σ−1(k), then

where Sck is the image of a fixed trivial string link, as in Definition 5.1.
As in the splicing operad, we think of σ∈Σn as a permutation in Σn+1=Aut{0,1,...,n} which fixes 0.

The space I(c1,...,cn;c) is the space of equivalence classes [L0,...,Ln,σ] of infection diagrams, where (L0,...,Ln,σ) and (L′0,...,L′n,σ′) are equivalent if Li=L′i for all i, and if whenever the images of Li and Lk intersect, σ−1(i)≤σ−1(k) if and only if (σ′)−1(i)≤(σ′)−1(k).
∎

L0L4L1L2L3L5L4(S3)L1(S1)Figure 7: An infection diagram, or more precisely, an element of
I(1,2,2,3,1;3).

Informally, the Li are like the hockey pucks in Budney’s splicing operad, and the permutation σ is a map that sends the order of composition to the index i of Li. The difference is that instead of re-embedding a hockey puck into itself, we will re-embed the image of Sci, a subspace of thinner inner cylinders, into the puck. Thus we keep track of the image of Sci, and our pucks can be thought of as having cylindrical holes drilled in them, the holes with which we will grab the string link (or long knot) L0. Following a suggestion of V. Krushkal, we call the restrictions of the Li to (I×D2)∖∘Sci “mufflers” (motivated by the picture for ci=2).

The generalization of Budney’s continuity constraint to the constraint (†) is the key technical ingredient in defining our colored operad. The need for this constraint is explained precisely in Remark 5.4 below. The rough meaning of this condition is that a muffler which acts earlier should be inside a hole of a muffler that acts later; in other words, the “solid part” of a higher Lk (which remains after drilling out the trivial string link) should not intersect any part of a lower Li, where “higher” and “lower” are in the semi-linear ordering determined by σ. However, we must allow for the possibility of the boundaries of the mufflers intersecting in certain ways. Figure 8 displays the cross-section of a set of mufflers which satisfy constraint (†).

Figure 8: The cross-section of a set of thirteen mufflers, including seven one-holed mufflers (or hockey pucks), satisfying the constraint (†). Each grey area is the “forbidden region” Lk(∘I×(D2∖∘Sck)) of the kth muffler, i.e., the region where no other muffler may lie.

So far we haven’t finished defining the operad, since we haven’t defined the structure maps. We start by defining the action on the space of fat string links. Only after that will we define the structure maps and check that they form a colored operad and that the definition below is a colored operad action.

Definition 5.3 (The action of I on fat string links).

Consider [L0,L1...,Ln,σ]∈I(c1,...,cn;c) and fat string links f1,...,fn where fk∈FSLck. Let Link be the map obtained from Lk by restricting the domain to Sck and restricting the codomain to its image. We use the shorthand notation

Lk⋅fk to denote the map Lk∘fk∘(Link)−1:Lk(Sck)→I×D2.

Then we define

I(c1,...,cn;c)×FSLc1×...×FSLcn

→FSLc

by

([L0,L1,...,Ln,σ],f1,...,fn)

↦(Lσ(n)⋅fσ(n))∘...∘(Lσ(1)⋅fσ(1))∘L0.\qed

f1f2f3L0L1L3L2(L3⋅f3)∘...∘(L1⋅f1)∘L0L1⋅f1L3⋅f3L2⋅f2Figure 9: The result of an element of an infection diagram acting on three fat string links, or more precisely a map I(1,3,2;2)×FSL1×FSL2×FSL3→FSL2. The 2-component fat string link at the bottom is the result of this action.

Remark 5.4.

Strictly speaking, each map Lσ(k)⋅fσ(k) is only defined on Lσ(k)(Scσ(k))=imLinσ(k), so one might worry whether the above composition is well defined. We claim that the conditions on the support of the fσ(k) and the continuity constraint (†) guarantee that we can continuously extend each Lσ(k)⋅fσ(k) by the identity on imL0∖imLinσ(k).

In fact, first write

∂(imLinσ(k))=(∂I×∐ckD2)∪(I×∂∐ckD2).

Since each fσ(k) is the identity on the ∂I×∐ckD2 part of its domain (the “flat boundary”), the map Lσ(k)⋅fσ(k) is the identity on the ∂I×∐ckD2 part of imLinσ(k).